I was working with my five-year-old in Dimensions math, and we came across an exercise that asks the students to circle all of the circles shown. Some of the shapes represent cylinders; one represents a football. Obviously the top and bottom of a cylinder are circles. But what about a football? This led to an interesting discussion about dimensions.

I asked my son to imagine a football and then imagine that the point was toward his face. What would he see? A circle! So the cross-section of the football is a circle, so he circled the football. (I added in a little note with the relevant qualifier.)

Then we had some even deeper discussions on the topic. Is the ceiling 2-dimensional or 3-dimensional? It’s flat—mostly. But we have a popcorn ceiling, so it’s obvious the ceiling is not perfectly flat. What about paper? At that point we actually got out the microscope to look at paper magnified. I pulled up some good imagery of paper shown under really good microscopes. Obviously paper is not truly 2-dimensional.

There was a funny moment where I explained (as best I could) that nothing in real life actually is 2-dimensional; everything is 3-dimensional. A 2-dimensional plane is a mental abstraction in which we imagine away height. My son was (momentarily) crestfallen that 2-dimensional things don’t actually exist (beyond our abstractions). But, for practical purposes, we can take certain things as flat, as they’re flat enough to assume flatness for the task at hand. (More technically, I’d say the concept of “flat” does not imply a lack of small-scale 3-dimensionality).

Then we also talked about the difference between a 2-dimensional shape as printed on a flat piece of paper versus the 3-dimensional object that such a drawing (sometimes) represents. To drive home this point I held up a real cylinder so we could see how its shapes appear from our momentary perspective. As printed, the “cylinders” on the paper are drawn using ovals, not circles. But those ovals represent circles of a cylinder.

This was pretty heavy-duty stuff for a five-year-old. He seemed to basically follow the discussion—although I’m sure we’ll have to come back to those difficult ideas many times before they fully “stick.”

Here is the broader pedagogical point: I never assume that the most important lesson is the one most obviously at hand. My kid knows what a circle is; we were just quickly pushing through the book at hand as review. But dimensionality—now that’s interesting!